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Categorial aspects of topological groups and rings theory


Author: Alina Alb Daciana
Degree:doctor of physics and mathematics
Speciality: 01.01.06 - Mathematical logics, algebra and theory of numbers
Year:2005
Scientific adviser: Mihail Ursul
doctor habilitat, Tiraspol State University
Scientific consultant: Mitrofan Cioban
doctor habilitat, professor, Tiraspol State University
Institution:
Scientific council:

Status

The thesis was presented on the 28 March, 2005
Approved by NCAA on the 23 June, 2005

Abstract

Adobe PDF document0.29 Mb / in romanian

Thesis

CZU 512.556

Adobe PDF document 0.77 Mb / in romanian
78 pages


Keywords

epireflective subcategory, coreflective subcategory, compact module, bounded ring, Jacobson radical, m-topology, variety of compact rings, flat compact module, tensor product of compact modules (rings), semisimple ring in the sense of Jacobson, coproduct of topological Abelian groups, countably complete topological groups

Summary

In sixties of the last century there appeared a new direction in general topology, namely, categorical topology. The founders of this theory are H. Herrlich, J. Kennison, M. Husek and others.

We study in the thesis the following categorical aspects of the theory of topological modules and rings:

1. the characterization of epireflective subcategories of the category of all R-modules, where R is a topological ring with identity;

2. the characterization of epireflective subcategories of the category RC of all compact modules over a topological ring R with identity;

3. the characterization of simple subcategories of the category of all compact Abelian groups;

4. we construct different reflective subcategories of the category of all topological rings and prove the commutativity of the corresponding reflexion functors with topological products;

5. it is introduced a new cardinal invariant for all topological rings, using this cardinal are constructed coreflective subcategories of the category of all topological modules and is given a partial answer to a question of Professor M. Cioban concerning the coreflectivity of the category of P-modules;

6. it is introduced the notion of a variety of compact rings, is extended Birkhoff’s Theorem for the class of compact rings and is studied the problem of associativity of the groupoid of varieties of compact rings;

7. it is introduced the concept of a tensor product of compact modules. By using this notion it is given a characterization of compact semisimple rings in the sense of Jacobson.

The results of the thesis can be applied in the theory of topological modules, in the theory of topological rings, in category theory and in general topology.